■サマーヴィルの等面四面体(その431)

 n=2rのとき

  ρ/l=((2r+1)/8r(r+1))^1/2

をnに戻すと

  ρ/l=((n+1)/4n(n/2+1))^1/2

  ρ/l=((n+1)/2n(n+2))^1/2

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[1]nが偶数のとき

Σp^2(n−p+1)^2

=n(n+1)(n+2){n^2+2n+2}/60

が正しいとしたら,

 k^2・n(n+1)(n+2){n^2+2n+2}/60=(n+1)/2n(n+2)

 k^2=30/{n^2+2n+2}

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[2]nが奇数のとき

k^2・Σp^2(n−p+1)^2

=30/{n^2+2n+2}・(n+1)(n−1){8n^3+25n^2+28n+15}/480

=(n+1)(n−1){8n^3+25n^2+28n+15}/16{n^2+2n+2}

n=3を代入すると

  8・{216+225+84・15}/16・17

=8・540/16・17  (NG)

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